Nash-Williams [6] formulated a condition that is necessary and sufficient for a countable family A = (Ai)i∈I of sets to have a transversal. In [7] he proved that his criterion applies also when we allow the set I to be arbitrary and require only that ⋂ i∈J Ai = ∅ for any uncountable J ⊆ I. In this paper, we formulate another criterion of a similar nature, and prove that it is equivalent to the ...

(C1) ai,n = am,j = 0 for all non-zero i and j, (C2) ai,j = 0 for all pairs (i, j) satisfying ni+mj > mn, (C3) the sum of all monomials ai,jxy of F for which ni+mj = nm is a constant multiple of a power of an irreducible polynomial in Z[x, y]. We note that (C2) is a stronger condition than (C1). The reason that (C1) is included above will be made clear in the statement of Theorem 1. We will make...

Proof. As is known, a Bayesian network is equivalent to a topological ordering (Chapter 8, Section 8.1 on Page 362 in [1]). Therefore, we prove Proposition 1 by showing that i) Eqn. (1a ∼ 1d) lead to a topological ordering (the necessary condition), and ii) a topological ordering from a DAG can meet the requirements in Eqn. (1a ∼ 1d) (sufficient condition). First, we prove the necessary conditi...

This paper proposes a new method for estimating the error in the solution of linear systems. A condition number is defined for a linear function of the solution components. This definition of the condition number is quite versatile. It reduces to the component condition number proposed by Chandrasekaran and Ipsen [SIAM J. Matrix Anal. Appl., 16 (1995), pp. 93–112] and to Skeel’s definition of c...

When a strictly convex plane set S moves by translation, the set J of points of the integer lattice that lie in S changes. The number K of equivalence classes of sets J under lattice translations (configurations) is bounded in terms of the area of the Brunn-Minkowski difference set of S. If S satisfies the Triangle Condition, that no translate of S has three distinct lattice points in the bound...

Courtney J. Campbell, Allegra Giovine and Jennifer Keating’s edited collection brings together valuable research on emptiness as a historical condition. The overarching theme of is used a...

We present a uniied approach to subdivision algorithms for meshes of arbitrary topology admitting a rigorous analysis of the generated limit surfaces. A generalized suucient criterion for generating tangent plane continuous surfaces is provided. In particular, it applies to complex eigenvalues and also shows that the so-called double dominant eigenvalue condition, which was supposed to be indis...

begin{abstract} in this paper, we introduce an iterative method for amenable semigroup of non expansive mappings and infinite family of non expansive mappings in the frame work of hilbert spaces. we prove the strong convergence of the proposed iterative algorithm to the unique solution of a variational inequality, which is the optimality condition for a minimization problem. the results present...

suppose $pi:mathcal{a}rightarrow mathcal{b}$ is a surjective unital $ast$-homomorphism between c*-algebras $mathcal{a}$ and $mathcal{b}$, and $0leq aleq1$ with $ain  mathcal{a}$. we give a sufficient condition that ensures there is a proection $pin mathcal{a}$ such that $pi left( pright) =pi left( aright) $. an easy consequence is a result of [l. g. brown and g. k. pedersen, c*-algebras of real...

We study the strong asymptotics of orthogonal polynomials with respect to a measure of the type dμ/2π + ∑∞ j=1Ajδ(z− zk), where μ is a positive measure on the unit circle Γ satisfying the Szegö condition and {zj}j=1 are fixed points outside Γ. The masses {Aj}j=1 are positive numbers such that ∑∞ j=1Aj < +∞. Our main result is the explicit strong asymptotic formulas for the corresponding orthogo...